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The function given is $f(x) = [x]$, which is the greatest integer function. This function is defined as the greatest integer less than or equal to $x$.
The greatest integer function $[x]$ is known to be discontinuous at every integer value of $x$. For any integer $n$, the left-hand limit is $n-1$ and the right-hand limit is $n$. Since the limits are not equal, the function is discontinuous at all integers.
The given interval is $0 < x < 2$. Within this open interval, the only integer value present is $x = 1$.
Since $x = 1$ is the only integer in the interval $(0, 2)$, the function $f(x) = [x]$ is discontinuous at exactly one point within this interval.
Final Answer: At only one point